The Characteristic Function as a Unifying Framework for Linear Response in Surface Diffusion

In this short perspective, we analyze the different linear response functions relevant to surface diffusion as studied by helium atom scattering, organizing them around a single object: the intermediate scattering function (ISF), which is also a characteristic function (CF) in the sense of probability theory. This organizing role of the CF is, to our knowledge, not made explicit elsewhere in the surface-diffusion literature. The exponential time dependence of the ISF observed in the diffusive regime (times much greater than the inverse of the friction coefficient) is a special case of the classical continuous-time random walk (CTRW) theory. Special emphasis is placed on this regime where quantum features of the diffusion process are washed out. We show how the entire hierarchy of response functions—the after-effect function, the generalized susceptibility, the relaxation function, and the Green function—can be written directly in terms of the time moments of the ISF at t=0. Moreover, the standard Pauli master equation and the Chudley–Elliott (CE) jump model follow as particular lattice realizations of a general compound-Poisson process. The extension to finite surface coverage is discussed within the interacting single adsorbate (ISA) model.